Artificial Intelligence Propositional Logic Marc Toussaint University of Stuttgart Winter 2015/16 (slides based on Stuart Russell s AI course)
Outline Knowledge-based agents Wumpus world Logic in general models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution 2/64
Knowledge bases agent y 0 a 0 y 1 a 1 y 2 a 2 y 3 a 3 s 0 s 1 s 2 s 3 An agent maintains a knowledge base Knowledge base = set of sentences of a formal language 3/64
Wumpus World description Performance measure gold +1000, death -1000-1 per step, -10 for using the arrow Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it The wumpus kills you if in the same square Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square Actuators Left turn, Right turn, Forward, Grab, Release, Shoot, Climb Sensors Breeze, Glitter, Stench, Bump, Scream 4/64
Exploring a wumpus world 5/64
Exploring a wumpus world 6/64
Exploring a wumpus world 7/64
Exploring a wumpus world 8/64
Exploring a wumpus world 9/64
Exploring a wumpus world 10/64
Exploring a wumpus world 11/64
Exploring a wumpus world 12/64
Other tight spots Breeze in (1,2) and (2,1) no safe actions Assuming pits uniformly distributed, (2,2) has pit w/ prob 086, vs 031 Smell in (1,1) cannot move Can use a strategy of coercion: shoot straight ahead wumpus was there dead safe wumpus wasn t there safe 13/64
Logic in general Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language Semantics define the meaning of sentences; ie, define truth of a sentence in a world Eg, the language of arithmetic x + 2 y is a sentence; x2 + y > is not a sentence x + 2 y is true iff the number x + 2 is no less than the number y x + 2 y is true in a world where x = 7, y = 1 x + 2 y is false in a world where x = 0, y = 6 14/64
Entailment Entailment means that one thing follows from another: KB = α Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true Eg, the KB containing the Giants won and the Reds won entails Either the Giants won or the Reds won Eg, x + y = 4 entails 4 = x + y Entailment is a relationship between sentences (ie, syntax) that is based on semantics 15/64
Models Given a logical sentence, when is its truth uniquely defined in a world? Logicians typically think in terms of models, which are formally structured worlds (eg, full abstract description of a world, configuration of all variables, world state) with respect to which truth can uniquely be evaluated We say m is a model of a sentence α if α is true in m M(α) is the set of all models of α Then KB = α if and only if M(KB) M(α) Eg KB = Giants won and Reds won α = Giants won 16/64
Entailment in the wumpus world Situation after detecting nothing in [1,1], moving right, breeze in [2,1] Consider possible models for?s assuming only pits 3 Boolean choices 8 possible models 17/64
Wumpus models 18/64
Wumpus models KB = wumpus-world rules + observations 19/64
Wumpus models KB = wumpus-world rules + observations α 1 = [1,2] is safe, KB = α 1, proved by model checking 20/64
Wumpus models KB = wumpus-world rules + observations α 2 = [2,2] is safe, KB = α 2 21/64
Inference Inference in the general sense means: Given some pieces of information (prior, observed variabes, knowledge base) what is the implication (the implied information, the posterior) on other things (non-observed variables, sentence) KB i α = sentence α can be derived from KB by procedure i Consequences of KB are a haystack; α is a needle Entailment = needle in haystack; inference = finding it Soundness: i is sound if whenever KB i α, it is also true that KB = α Completeness: i is complete if whenever KB = α, it is also true that KB i α Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure That is, the procedure will answer any question whose answer follows from what is known by the KB 22/64
Propositional logic: Syntax Propositional logic is the simplest logic illustrates basic ideas The proposition symbols P 1, P 2 etc are sentences If S is a sentence, S is a sentence (negation) If S 1 and S 2 are sentences, S 1 S 2 is a sentence (conjunction) If S 1 and S 2 are sentences, S 1 S 2 is a sentence (disjunction) If S 1 and S 2 are sentences, S 1 S 2 is a sentence (implication) If S 1 and S 2 are sentences, S 1 S 2 is a sentence (biconditional) 23/64
Propositional logic: Syntax grammar sentence atomic sentence complex sentence atomic sentence true false P Q R complex sentence sentence ( sentence sentence ) ( sentence sentence ) ( sentence sentence ) ( sentence sentence ) 24/64
Propositional logic: Semantics Each model specifies true/false for each proposition symbol Eg P 1,2 P 2,2 P 3,1 true true false (With these symbols, 8 possible models, can be enumerated automatically) Rules for evaluating truth with respect to a model m: S is true iff S is false S 1 S 2 is true iff S 1 is true and S 2 is true S 1 S 2 is true iff S 1 is true or S 2 is true S 1 S 2 is true iff S 1 is false or S 2 is true ie, is false iff S 1 is true and S 2 is false S 1 S 2 is true iff S 1 S 2 is true and S 2 S 1 is true Simple recursive process evaluates an arbitrary sentence, eg, P 1,2 (P 2,2 P 3,1 ) = true (false true) = true true = true 25/64
Truth tables for connectives P Q P P Q P Q P Q P Q false false true false false true true false true true false true true false true false false false true false false true true false true true true true 26/64
Wumpus world sentences Let P i,j be true if there is a pit in [i, j] Let B i,j be true if there is a breeze in [i, j] P 1,1 B 1,1 B 2,1 Pits cause breezes in adjacent squares 27/64
Wumpus world sentences Let P i,j be true if there is a pit in [i, j] Let B i,j be true if there is a breeze in [i, j] P 1,1 B 1,1 B 2,1 Pits cause breezes in adjacent squares B 1,1 (P 1,2 P 2,1 ) B 2,1 (P 1,1 P 2,2 P 3,1 ) A square is breezy if and only if there is an adjacent pit 28/64
Truth tables for inference B 1,1 B 2,1 P 1,1 P 1,2 P 2,1 P 2,2 P 3,1 R 1 R 2 R 3 R 4 R 5 KB false false false false false false false true true true true false false false false false false false false true true true false true false false false true false false false false false true true false true true false false true false false false false true true true true true true true false true false false false true false true true true true true true false true false false false true true true true true true true true false true false false true false false true false false true true false true true true true true true true false true true false true false Enumerate rows (different assignments to symbols), if KB is true in row, check that α is too 29/64
Inference by enumeration Depth-first enumeration of all models is sound and complete function TT-Entails?(KB, α) returns true or false inputs: KB, the knowledge base, a sentence in propositional logic α, the query, a sentence in propositional logic symbols a list of the proposition symbols in KB and α return TT-Check-All(KB, α, symbols, [ ]) function TT-Check-All(KB, α, symbols, model) returns true or false if Empty?(symbols) then if PL-True?(KB, model) then return PL-True?(α, model) else return true else do P First(symbols); rest Rest(symbols) return TT-Check-All(KB, α, rest, Extend(P, true, model)) and TT-Check-All(KB, α, rest, Extend(P, false, model)) O(2 n ) for n symbols 30/64
Logical equivalence Two sentences are logically equivalent iff true in same models: α β if and only if α = β and β = α (α β) (β α) commutativity of (α β) (β α) commutativity of ((α β) γ) (α (β γ)) associativity of ((α β) γ) (α (β γ)) associativity of ( α) α double-negation elimination (α β) ( β α) contraposition (α β) ( α β) implication elimination (α β) ((α β) (β α)) biconditional elimination (α β) ( α β) De Morgan (α β) ( α β) De Morgan (α (β γ)) ((α β) (α γ)) distributivity of over (α (β γ)) ((α β) (α γ)) distributivity of over 31/64
Validity and satisfiability A sentence is valid if it is true in all models, eg, true, A A, A A, (A (A B)) B Validity is connected to inference via the Deduction Theorem: KB = α if and only if (KB α) is valid A sentence is satisfiable if it is true in some model eg, A B, C A sentence is unsatisfiable if it is true in no models eg, A A Satisfiability is connected to inference via the following: KB = α if and only if (KB α) is unsatisfiable ie, prove α by reductio ad absurdum 32/64
Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules Legitimate (sound) generation of new sentences from old Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search alg Typically require translation of sentences into a normal form Model checking truth table enumeration (always exponential in n) improved backtracking, eg, Davis Putnam Logemann Loveland (see book) heuristic search in model space (sound but incomplete) eg, min-conflicts-like hill-climbing algorithms 33/64
Forward and backward chaining Applicable when KB is in Horn Form Horn Form (restricted) KB = conjunction of Horn clauses Horn clause = proposition symbol; or (conjunction of symbols) symbol Eg, C (B A) (C D B) Modus Ponens (for Horn Form): complete for Horn KBs α 1,, α n, α 1 α n β β Can be used with forward chaining or backward chaining These algorithms are very natural and run in linear time 34/64
Forward chaining Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found P Q L M P B L M A P L A B L A B 35/64
Forward chaining example 36/64
Forward chaining example 37/64
Forward chaining example 38/64
Forward chaining example 39/64
Forward chaining example 40/64
Forward chaining example 41/64
Forward chaining example 42/64
Forward chaining example 43/64
Forward chaining algorithm function PL-FC-Entails?(KB, q) returns true or false inputs: KB, the knowledge base, a set of propositional Horn clauses q, the query, a proposition symbol local variables: count, a table, indexed by clause, initially the number of premises inferred, a table, indexed by symbol, each entry initially false agenda, a list of symbols, initially the symbols known in KB while agenda is not empty do p Pop(agenda) unless inferred[p] do inferred[p] true for each Horn clause c in whose premise p appears do decrement count[c] if count[c] = 0 then do if Head[c] = q then return true Push(Head[c], agenda) return false 44/64
Proof of completeness FC derives every atomic sentence that is entailed by KB 1 FC reaches a fixed point where no new atomic sentences are derived 2 Consider the final state as a model m, assigning true/false to symbols 3 Every clause in the original KB is true in m Proof: Suppose a clause a 1 a k b is false in m Then a 1 a k is true in m and b is false in m Therefore the algorithm has not reached a fixed point! 4 Hence m is a model of KB 5 If KB = q, q is true in every model of KB, including m General idea: construct any model of KB by sound inference, check α 45/64
Backward chaining Idea: work backwards from the query q: to prove q by BC, check if q is known already, or prove by BC all premises of some rule concluding q Avoid loops: check if new subgoal is already on the goal stack Avoid repeated work: check if new subgoal 1) has already been proved true, or 2) has already failed 46/64
Backward chaining example 47/64
Backward chaining example 48/64
Backward chaining example 49/64
Backward chaining example 50/64
Backward chaining example 51/64
Backward chaining example 52/64
Backward chaining example 53/64
Backward chaining example 54/64
Backward chaining example 55/64
Backward chaining example 56/64
Forward vs backward chaining FC is data-driven, cf automatic, unconscious processing, eg, object recognition, routine decisions May do lots of work that is irrelevant to the goal BC is goal-driven, appropriate for problem-solving, eg, Where are my keys? How do I get into a PhD program? Complexity of BC can be much less than linear in size of KB 57/64
Resolution Conjunctive Normal Form (CNF universal) conjunction of disjunctions of literals }{{} clauses Eg, (A B) (B C D) Resolution inference rule (for CNF): complete for propositional logic l 1 l k, m 1 m n l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n where l i and m j are complementary literals Eg, P 1,3 P 2,2, P 2,2 P 1,3 Resolution is sound and complete for propositional logic 58/64
Conversion to CNF B 1,1 (P 1,2 P 2,1 ) 1 Eliminate, replacing α β with (α β) (β α) (B 1,1 (P 1,2 P 2,1 )) ((P 1,2 P 2,1 ) B 1,1 ) 2 Eliminate, replacing α β with α β ( B 1,1 P 1,2 P 2,1 ) ( (P 1,2 P 2,1 ) B 1,1 ) 3 Move inwards using de Morgan s rules and double-negation: ( B 1,1 P 1,2 P 2,1 ) (( P 1,2 P 2,1 ) B 1,1 ) 4 Apply distributivity law ( over ) and flatten: ( B 1,1 P 1,2 P 2,1 ) ( P 1,2 B 1,1 ) ( P 2,1 B 1,1 ) 59/64
Resolution algorithm Proof by contradiction, ie, show KB α unsatisfiable function PL-Resolution(KB, α) returns true or false inputs: KB, the knowledge base, a sentence in propositional logic α, the query, a sentence in propositional logic clauses the set of clauses in the CNF representation of KB α new { } loop do for each C i, C j in clauses do resolvents PL-Resolve(C i, C j ) if resolvents contains the empty clause then return true new new resolvents if new clauses then return false clauses clauses new 60/64
Resolution example KB = (B 1,1 (P 1,2 P 2,1 )) B 1,1 α = P 1,2 61/64
Summary Logical agents apply inference to a knowledge base to derive new information and make decisions Basic concepts of logic: syntax: formal structure of sentences semantics: truth of sentences wrt models entailment: necessary truth of one sentence given another inference: deriving sentences from other sentences soundness: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences Wumpus world requires the ability to represent partial and negated information, reason by cases, etc Forward, backward chaining are linear-time, complete for Horn clauses Resolution is complete for propositional logic Propositional logic lacks expressive power 62/64
Dictionary: logic in general a logic: a language, elements α are sentences, (grammar example: slide 34) model m: a world/state description that allows us to evaluate α(m) {true, false} uniquely for any sentence α, M(α) = {m : α(m) = true} entailment α = β: M(α) M(β), m : α(m) β(m) (Folgerung) equivalence α β: iff (α = β and β = α) KB: a set of sentences inference procedure i can infer α from KB: KB i α soundness of i: KB i α implies KB = α (Korrektheit) completeness of i: KB = α implies KB i α 63/64
Dictionary: propositional logic conjunction: α β, disjunction: α β, negation: α implication: α β α β, biconditional: α β (α β) (β α) Note: = and are statements about sentences in a logic; and are symbols in the grammar of propositional logic α valid: true for any model, eg: KB = α iff [(KB α) is valid] (allgemeingültig) α unsatisfiable: true for no model, eg: KB = α iff [(KB α) is unsatisfiable] literal: A or A, clause: disjunction of literals, CNF: conjunction of clauses Horn clause: symbol (conjunction of symbols symbol), Horn form: conjunction of Horn clauses Modus Ponens rule: complete for Horn KBs α1,,αn, α1 αn β β Resolution rule: complete for propositional logic in CNF, let l i = m j : l 1 l i 1 l m j+1 m n l 1 l k, m 1 m n i+1 l k 1 m j 1 m 64/64